If I did live in Antartica it would have to be valid wouldn't it?

Your argument is that: If logicians have defined validity, then that definition is correct. Logicians have defined validity. Therefore, that definition is correct. This is a valid argument as far as I can tell. It is, however, unsound, as premise 1 is faulty.

Besides, if someone gave the argument you gave -- "I am a man and I am not a man. Therefore I am rich" that is a nonsensical argument; the conclusion just has nothing to do with the premises, you might as well argue "I am a human and it might snow this week, therefore I live in Antartica." Even if conclusion and premise are all true i.e. the argument is sound, what kind of argument is that?

It seems that that argument would be valid, but only if one accepts that an argument is valid iff there is no interpretation s.t. all premises are true and the conclusion is false per Tones' definition.

If it turned out that validity required more than what that definition suggests (I think it does), then the argument you stated may well turn out to not be valid, as I think is the case.

Maybe another way of coming at this is as follows - the conclusion is true. Period. Under that understanding, "there is no interpretation where the conclusion is false" ergo there is no interpretation s.t. all the premises are true and the conclusion is false. But the conclusion being true does not seem to guarantee that the argument is valid. But with Tones' definition, it would. Similarly, inconsistent premises also guarantee the validity of the argument according to Tones' definition, but that also seems problematic.

"Validity has to do with the conclusion following from the premises, and inconsistency is not evidence that the conclusion follows from the premises." — Leontiskos

That ((P→Q)∧Q), therefore P is not valid, whereas ((A∧¬A)∧(P→Q)∧Q), therefore P is valid, does seem strange to me. Inconsistent premises don't seem to have anything to do with whether the argument "follows." Although I have a feeling that Tones will have something to say about that.

One of the main takeaways from this discussion, for me, is that while some formal arguments may be valid, they are not necessarily valid in an informal setting.

To wit,

B
Therefore A→B
Formally valid.

Water was added to the lake.
Therefore,
If it is cloudy out, then water was added to the lake.
Informally not valid.

as well as -

A ^ B
Therefore, (A→B).
Formally valid.

Kangaroos are marsupials and Paris is the capital of France.
Therefore,
If kangaroos are marsupials, then Paris is the capital of France.
Informally not valid.

If I am referring to the right quotation, you said:

No, it doesn't result in a contradiction. The conclusion is ~A, which is not a contradiction. Yes, the premises are inconsistent, but your definition of "rule" doesn't disallow inconsistent sets of premises, only required is that application of the rule doesn't allow a conclusion that is a contradiction. The particular application you mentioned doesn't derive a contradiction. — TonesInDeepFreeze

What I responded with --a rule must have been "followed" not merely be "present" and the use of a rule may not result in a contradiction means that the use of a rule, or I guess you would call it an operator or connective, whatever you call it, must not result in a contradiction. A->not-A, when this rule is applied and followed, that is, when it is true that "A" and the rule "A->not-A" is actually applied, a contradiction results, specifically "A and not-A."

By "actually applied" I mean that the rule, or connective, does work in leading to the conclusion.

The "following" of a rule versus it's being merely "present" can be illustrated by the following example:
A->B
B^C
Therefore, C.
In this example, the rule A-> B does not do any work, so even if it did result in a contradiction, the fact that it doesn't do any work in the argument and isn't followed or actually applied, means that the argument could still be valid.

Down the slippery slope of formalized illogicality.

You're slipping Tones.

Then note:

P -> Q |= ~P v Q
and
~P v Q |= ~P v Q — TonesInDeepFreeze

I think you meant:

P -> Q |= ~P v Q
and
~P v Q |= P -> Q

?

Not just conjunction, no, but having the same truth functionality as conjunction yes, just meta-logically different (if I am using that terminology correctly).

No, I read it, I just think you're disregarding the proviso I stated, namely that a rule must actually have been followed, not merely be present in an argument.

As for the instantiation of truth possibilities by the rules, what I mean is that the possibilities for what is true and what is false are arrayed across a truth table. The rules must account for all the ways that those truth possibilities can be instantiated. So for the expression A v B, the truth table is T, T, T, F. On the other hand, T, F, F, F, is A ^ B. Every possibility wherein T is present must be uniquely accounted for by the rules. So T, F, F, F, and F, T, F, F, and F, F, T, F, and F, F, F, T, must all be "achievable instantiations" based on the rules we bring to the variables. If A v B were the only rule we applied, then not all of the truth possibilities could be instantiated, does that answer?

By "the following of a rule" I mean a literal rule such as a connective is actually used to reach a conclusion. The argument A->not-A therefore not-A does not, in my opinion, make any use of the conditional such that any rule has been followed. With the argument A->not-A, A, therefore not-A, the following of the rule, namely the conditional in that argument, leads to a contradiction between A and not-A, as such, it is disqualified from being a valid argument according to my definition.

According to you, what is the full meaning of P -> Q? — TonesInDeepFreeze

I may have mispoken, but to me the full meaning of "If P then Q" captures the fact that "P does not imply Q" can still be true even though not-P v Q can still be true. But then I now think P->Q is a meaningless expression so saying it "means" the same think as not-P or Q is unsubstantiated.

Whether or not the two expressions are semantically equivalent in a meta-logical sense depends on how one is using them. — Leontiskos

Hmm interesting, I think my position is that the formal conditional is meaningless then, insofar as it is just symbol manipulation.

You could say that, but you would end up having to admit that "P does not imply Q" cannot be formalized in any way whatsoever, at least in propositional logic. — Leontiskos

I have tried to formalize it and can't seem to do so; this is an approximation:

(A v ~A) → (~B v ~A)

When (B and A) are both true, the expression seems to be false. On the other hand, the negation of that expression seems to imply that (A and B) must both be true. If the conditional is construed as only being true when A and B are true, then the negation of the initial expression maps onto A→B. Perhaps that could be written as, it is not the case that A does not imply B therefore A implies B. (Though if that were the case then A→B would be logically equivalent to A^B, although not meta-logically equivalent).

But then I don't mind saying "P does not Imply Q" can't be formalized.

I get mixed up with this, but I think the disjunction (not-P or Q) can still be true even if P does not imply Q. So the "meaning" of the disjunctive is not specific enough.

It seems to me that the disjunctive equivalent does not capture the full meaning of P->Q.

You can absolutely substitute them logically, however I do not think they mean the same thing. P->Q either means just that "P->Q" or it doesn't have a meaning at all, either way P->Q does not, in my opinion, mean the same thing as its logical equivalent.

1. Right, I mean P entails Q. The logical equivalence (not-P or Q) is an implication of the conditional, not having the same meaning as the conditional.

2. I take your question to be what would a rule be, how is it defined? I would define a rule as a member belonging to a set that exhausts all "truth possibilities." I would add that the following of a rule may not result in a contradiction.

A rule relating two different variables would have (I think) 15 possible truth configurations. The rules must at least enable all those possibilities to be instantiated (though perhaps it may exclude possibilities that are necessarily contradictory).

3. "Some proposition is not the case"
Both propositions must be true
Either proposition must be true
If the one proposition is true, so must the consequent proposition
Both propositions are either both true or both false.

5. Valid argument = following the rules, where rules are defined as those operations that enable each truth possibility to be instantiated but that do not result in a contradiction by following that rule.

8. Not logical anarchy; the rules must enable all truth possibilities to be instantiated except that the rule may not result in a contradiction if it is followed.

This way of defining validity may be preferable because it deals with cases such as A->not-A therefore Not-A that are intuitively illogical; such an argument does not involve the following of a rule, and so it is not valid.

Similarly, A, A->not-A therefore not-A another intuitively illogical seeming argument would not be valid because the following of the rule results in a contradiction.

1. I take a conditional to be saying: if the antecedent is true, it can't be the case (there is no circumstances such) that the consequent is false.

2. Rather than a correct conclusion, all we need are conclusions that follow the relevant rules, any and all such conclusions are legitimate.

3. I refer to connectives as rules.

4. Then we are out of luck.

5. I drop the truth preservation condition for validity.

8. If we drop the truth preservation part of the definition, it is not circular. An argument is valid where it follows the relevant rules. Period. I don't think it is necessary for me to stipulate that a rule be followed "correctly," just that it be followed.

1. The conditional means that in the event that the antecedent is true, the consequent must be true. It is one of the logical rules that must be followed for the argument to be valid.

2. Provided that a set of all conclusions follows the rules correctly and is exhaustive of all such conclusions, that set encompasses all legitimate conclusions.

3. logical operators.

4. We would have to ask the speaker to clarify.

5. Noted, let's set aside questions concerning meaning; the second definition may have more problems then I can resolve.

6. Okay.

7. So then "the truth of the premises guarantees the truth of the conclusion" is the same as "there is no interpretation (assignment of truth values) such that the premises are all true and the conclusion is false" ?

8. So I think what I am trying to say is that the definition of validity is following the rules correctly. And that following the rules correctly is defined by rule-following that results in truth preservation. Such that, truth preservation is a consequence of rule following, and it is the rule following itself that is responsible for the validity. In other words, the premises themselves don't guarantee the truth of the conclusion, rather the following of the rule(s), given that the premises are all true, is what guarantees the truth of the conclusion. Put another way, truth preservation does not make the argument rule-following, but rule-following is what makes the argument truth preserving. (Truth preservation does make the rule-following "correct.") Not sure if that totally makes sense.

So actually, I would say my definition of valid is different from the ordinary formal logic definition in that I am defining validity in terms of rule-following, not in terms of truth-preservation; truth-preservation is more like a consequence of the definition.

P->Q. P. Therefore, not-Q. would both flout the meaning of the conditional, and in such a way that it changes the conclusion. It's different than what the conclusion should be (namely Q).

I don't understand the second question.

Third question answered as correctly used rules is defined.

I don't know the difference between propositional logic and ordinary formal logic so I do not know how to answer this one.

The meaning of an expression depends on what the speaker intended by it - natural language I would think would go along way in dissolving confusion over what is meant.

Right, where there is disagreement over a meaning, that meaning is not well-formed and not suitable for logical operations. I would expect something like that to be true for your definition of validity as well.

I guess I agree with the ordinary definition of valid in formal logic. That is not the definition you cited earlier in the thread - the definition that I am suggesting an alternative to.

I do not see truth preservation as synonymous with validity; I defined validity as rule following; a rule is followed correctly if it preserves truth; I didn't define validity as truth-preserving. Truth preservation is a consequence of validity, namely, following the relevant rules correctly.

And a relevant rule is correctly followed just in case.. if it were the case that all the premises were true and the relevant rule is followed, then the conclusion must also be true.

Relevant rules like conditionals "And" "Or" operators-- when those are used correctly the rules are followed and the argument may be considered valid. Any rule that is such that if it weren't followed, the conclusion would be different, is a relevant rule. The rules would ideally be universal and based on logical intuition; if people use different sets of rules, then the rules must be clearly communicated so that that "logic" can be understood or followed.

The meaning of the premise and conclusion depends on the expressions used (I guess this definition isn't unequivocal as it would only apply to ordinary natural language, not to formal logic). I don't know any theories of meaning so I can't answer that. If the meanings differ, then I'm not really sure what the result would be, seems like communication is out the door let alone logic if we can't agree on the same meaning of words and sentences.

Here are two ideas for defining validity: (1) an argument is valid when all the relevant rules are followed. Or, (2) an argument is valid when the meaning of the premises leads to the meaning of the conclusion.

It has been a long time since I learned some logic and I wasn't great at it, but I do know what truth tables are and I think how to use them.; I don't see how that implies a definition of "validity" using classical logic.

I mean take the definition of validity, and write it as an expression using symbols and logical operators; is that something that can be done?

I don't mean examples of valid arguments, I am referring to the definition itself.

Okay, I actually do get that the example I just gave has "an interpretation wherein all the premises are true and the conclusion is false" such that it is "not valid." " Would you care to formalize the validity definition as it concerns arguments and do so using logical operators? I was trying to apply De Morgan's laws to your definition but I don't think it worked. On a side note, Banno I can hear your laughter and it is most unwelcome at this time.

Or even if just one (but not all) of the premises is false and the conclusion is false (I am having trouble thinking of an example that meets this description).

A -> ~A makes sense whether A is true or A is false. — TonesInDeepFreeze

I am not clear on how A -> not-A "makes sense" if A is true.

Also, TonesInDeepFreeze, an argument where all the premises are false and the conclusion is false would necessarily be valid; is that correct?

I was thinking of:

P->not-Q
not-P
Therefore,
not-Q.

Assuming that all the premises are false and the conclusion is false, the argument must be valid. Is that correct?

Okay, correct me if I'm wrong, but you are saying that ordinary natural language is "mappable" onto formal classical logic because in formal logic a syntactic inconsistency viz., a negated sign that is present alongside the original sign, results in an argument that is "not derivable" whether the sign and negated sign are explicitly present or present by implication (A->not-A). So just as the ordinary natural language argument is meaningless, so the classical logic argument is underivable.

Can an underivable argument be valid? (I suppose you would say "yes" because the "underived" (unconditioned) constituents of the argument are mere possibilities).

I would think many people would apply a truth table to the argument (A->notA therefore not-A), as I did, and see based on that, that the premise is only true when "not-A." Maybe "infer" is too strong a word for the conclusion of not-A.

The conclusion does not seem to "follow" or be a "logical" conclusion when we apply the argument to ordinary natural language.

So I guess what I'm wondering is whether an underivable (or meaningless) argument may be regarded as logical? Or are soundness and validity insufficient for a logical argument? Or is meaning related to soundness?

It seems, to me, as though what is meant is critical to determining whether an argument is logical.

While you are proving what exactly is logical, you might as well prove that 2+2=4 and that there is an external world, but I don't want to hear any of that mathematical intuition or logical intuition or perceptual intuition nonsense.

Then I challenge you to prove that the following argument is logical:

P
P->Q
Therefore Q.

Or that this argument is logical:

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

Or that this argument is logical:

If it rained yesterday then the lake is swollen today.
It rained yesterday.
Therefore, the lake is swollen today.

Well it seems to me that all we can rely on when it comes to logic is intuition. If logic is just a formal set of rules as to how symbols may relate then anything can be logical and in that case nothing really is "logical," though I take that to be the discussion in the logical nihilism thread.

Our logical intuitions are basic, or foundational for doing logic, much in the way that having a functional ear is foundational for making a musical symphony.

One could argue that P->Q and P together implies not-Q, but translating that into natural language with the conditional spoken as an "if...then..." (or A and not-A therefore A and not-A) will be very difficult and I would say impossible, and that's because logic relies on meaning maybe just as much as meaning relies on logic.

All that to say that, at least informally A->notA therefore not-A may not be valid after all if our starting point is a set of meaningful natural language propositions.

That doesn't imply that formal logic is merely "academic" because it clearly has application to fields like computer science and mathematics.

But it may imply that some definitions we use in formal logic may be reviseable or at least more fungible then we previously thought.

↪NotAristotle
Is it worth pointing out, again, that "P→~P" is not a contradiction? If P→~P is true, then P is false.

If that's been said once, it's been said a thousand times... which is not once. — Banno

I know Banno; I am not disagreeing with the formal validity of that argument.

there are ambiguities in the English use of "If... then...", "...or..." and various other terms that we must settle in order to examine the structure of our utterances in detail. — Banno

I don't disagree with that either. But the argument A → ~A ∴ ~A clearly does not translate into natural language very well (I don't think there is any way to translate it in a way that renders the translation sensible and "logical"). And yet, the argument is valid formally speaking.

Michael suggested that the argument is not sound in ordinary language. I think he may be right. However, even arguments that are not sound can still be valid such that we can understand how the speaker reached their conclusion (though we may point out to them that such-and-such premise is not true). For example, if someone argued:

1. P
2. P→Q
Therefore, Q.

We might correct them, "well, actually ~Q." "Your reasoning is spot on and logical, it just happens to be that ~P, so while your reasoning is valid, the argument you presented is unsound."

On the other hand, "If it is raining, then it is not raining, therefore it is not raining" sounds like an unwarranted leap that is not logical when we consider it in an informal way. The problem isn't just that the initial premise is unsound (within an informal context); the problem is that the argument just doesn't make sense and is not logical, so soundness aside, that is why I call it "not valid" informally.

In fact, I would say A->B does not "mean" B or not-A.

I think you mean to say that the one implies the other through logical equivalence. That is different than saying that the expressions mean different things.

"I disagree with regards to ordinary language" I'm not quite getting it, what is the disagreement you have concerning ordinary language? You think someone would make an inference from A->not-A to therefore not-A in ordinary language?

Can you explain how those meanings diverge?